Exceptionally clean single-electron transistors from solutions of molecular graphene nanoribbons

Only single-electron transistors with a certain level of cleanliness, where all states can be properly accessed, can be used for quantum experiments. To reveal their exceptional properties, carbon nanomaterials need to be stripped down to a single element: graphene has been exfoliated into a single sheet, and carbon nanotubes can reveal their vibrational, spin and quantum coherence properties only after being suspended across trenches1–3. Molecular graphene nanoribbons4–6 now provide carbon nanostructures with single-atom precision but suffer from poor solubility, similar to carbon nanotubes. Here we demonstrate the massive enhancement of the solubility of graphene nanoribbons by edge functionalization, to yield ultra-clean transport devices with sharp single-electron features. Strong electron–vibron coupling leads to a prominent Franck–Condon blockade, and the atomic definition of the edges allows identifying the associated transverse bending mode. These results demonstrate how molecular graphene can yield exceptionally clean electronic devices directly from solution. The sharpness of the electronic features opens a path to the exploitation of spin and vibrational properties in atomically precise graphene nanostructures.

Then the reaction was stirred at 80 °C for 12 h. Afterwards, the reaction mixture was diluted with DCM (50 mL) and filtered. The mixture was washed three times with water, dried over sodium sulfate, and evaporated. The solid was purified by silica column chromatography

2,5-Bis(4-(anthracen-9-yl)phenyl)-3-phenyl-4-(3((triisopropylsilyl)ethynyl)phenyl)
cyclopenta-2,4-dien-1-one (S4             Figure S13c shows traces with clear current-steps as is typical for sequential electron tunnelling. Stability diagrams of the device showed Coulomb diamonds of many different sizes, which indicate the presence of multiple transport channels with different addition energies (from different MGNR lengths) and electrode coupling strengths (Figs.S15a,b,d). The red and the black diamond patterns in Figures S15a and S15b highlight Coulomb diamonds originating from two different MGNRs (among many) that bridges the nanogap. The gate lever arm, , for these two sets were estimated from the slopes of the diamonds as =0.015 and =0.007 for the red and the black sets, respectively. Their respective addition energies, add =150 meV and add =70 meV were extracted from the height of the diamonds. Figure S15c highlights the evolution of the I-VSD characteristics as a function of VG close to a degeneracy point. hosts 874 pairs of Cr (10 nm)/Au (70 nm) patterned by EBL lithography and metal evaporation.
The whole wafer was then sent to Graphenea for wafer-scale transferring of CVD monolayer graphene, with 600 nm of PMMA stacked on top of it. PMMA was then removed by leaving the chip in warm acetone overnight. Negative resist ARN-7500 was spun at 4000 rpm for 60 s, baked at 85 °C and exposed in EBL at 300 µC/cm 2 . 200 nm-wide graphene notches were then developed in MF-CD-26 for 60 s, immersed in deionised water for 60 s (no agitation), and blow-dried with N2.
Unwanted graphene features were removed by etching with a Henniker Oxygen Plasma (25 scmm, 50 % of power for 90 s). Afterwards, chips were immersed in REM-660 for one hour to remove cross-linked resist (agitation is not needed). The chips were rinsed in acetone for 1 min, followed by 1 min in IPA, and gently blow-dried with N2. Finally, nanogaps were formed using an electro-burning protocol and ~8 µL of a 1 mg/mL solution of 2 was drop casted onto the chip immediately after the formation of the junction.

Room-temperature Transport Properties of 2
Figure S16a shows the stability diagram for a device where 2 has been successfully deposited through drop casting, taken at room temperature in air. Figure S16b shows the tunnelling current of the break-junction after the electro-burning, the Simmons fit 3 (fitting paramaters: conductance. Therefore, looking for evenly spaced conductance peaks that run parallel to the diamond edge, and do not shift or split in a magnetic field, is one method of isolating vibrational states from the others. Caution is advised when using this method because not all electronic states (e.g. singlets) will split in a magnetic field and may be mistaken for a vibrational state.
Ab-initio calculations of the vibrational states should be used in combination to verify or suggest vibrational modes.
Visual inspection of the stability diagram in Figure 3d in the main text revealed conductance ridges evenly spaced by ~7.5 meV in all conducting regions that did not split with magnetic field. To differentiate between slanted conductance ridges from the leads, a script was made to average the conductance along lines running parallel to the diamond edges. In this way, only conductance ridges running approximately parallel to the diamond edges remained while slanted ridges were eliminated. The assumption is thus that the average conductance as a function of | | excludes the contribution of the lead transitions. The second assumption is that only peaks separated by ~7.5 meV represent the vibrational states of interest (thereby excluding electronic states). Figure S13a shows this procedure. For the phonon calculations, a k mesh grid was chosen to be 20 × 1 × 1 and energy cut-off 400 Ry. The dispersion relation is calculated with a supercell size of (3, 1, 1), where 3 represents 3 unit cells along the periodic direction. Atomic displacement of the Gamma phonon mode is visualized with XCrySDen software (Fig.4e in the main text). We need to point out that the side groups in our calculation are simplified. Since we have to compromise with the computational capability of DFT program, we kept only anthracene structure in the side group.  can estimate that the bending will be mostly confined at the backbone centre if we include the full side group structure into consideration.

Fitting the Conductance Peaks of Vibrational Origin. As explained in the main text,
Franck-Condon theory predicts that conductance peak intensities ,0 associated with a single vibron in equilibrium follow the progression given by Equation S1 : where is the vibron quantum number and is the electron-vibron coupling. The conductance, averaged along the diamond edge, as explained above, was plotted as a function of chemical potential, = | |Δ . The linear background of was subtracted eliminate other contributions than that of the single vibron, such as electronic excited states, density-of-states fluctuations and transport assisted by the environment-associated phonon bath.
,0 was fitted using Equation S1 assuming = 1, 2, 3, … 6, as fitting parameter and a peak energy spacing of = 7 meV. Figure S13b shows these fits for different in the range of 0.9 to 1.5. = 1.4 yield the best fit. > 1 implies strong electron-vibron coupling in the MGNR and is manifested by the first peak being lower than the second, i.e. 1,0 < 2,0 . This observation implies Franck-Condon suppression of SET though the ground state. We performed such Franck-Condon analyses on different charge states and similar values for were found, except for the transition − 3 ↔ − 2 where was slightly less than unity ( ≈ 0.8). However, our conductance peaks are best fitted with a Gaussian line shape, which is characteristic for transport limited by temperature-broadening. We note that the small number of measured conductance values for each peak makes it difficult to conclude whether the line shape is of Gaussian or Lorentzian type.

Line Width of Diamond Edge at Different Temperatures. The left edge of the N-3
diamond with a small positive bias ( ≈ −12.8 V) was analysed at different temperatures to investigate the effect of temperature on Franck-Condon suppression. The linewidth of the conduction peak at the diamond edge was estimated for the same diamond at three different temperatures, = 25 mK, 0.5 K and 1 K. A number of adjacent traces close to the degeneracy point were averaged and then fitted using the functions in Equations S2 and S3 (Fig.S14). Equation S2 is a Lorentzian that is proportional to lifetime-broadened conductance peaks: 7 where w is the FWHM of the conductance peak.
(S3) Figure S15a shows that Equation S2 fits the data well at all temperatures and that w increases linearly with T from w=0.63 meV at 25 mK to w=0.98 meV at 1 K. Figure S15b shows that the data also fits well to Equation 3 but with an extracted T that is too high, in particular at 25 mK where the electron temperature and the extracted value of T differs by two orders of magnitude. Therefore, we conclude that transport is dominated by lifetime broadening of the MGNR energy levels at low temperature where w is the coupling energy to the leads. We expect thermal broadening to become dominant at around 3 K. -deviation of Conductance Peak Intensities. We observed some dependency on the diamond-edge conductance-peak intensities. Figure S16 shows that of the first two peaks of a trace (analogous to the two first peaks in Figure S13b) depend on , or both.
This phenomenon is particularly visible at T = 0.5 K where the two peaks are approximately equal at low but shows an increasing difference up to about = 10 mV where it reaches a saturation of around 4 ⋅ 10 −5 / 0 . There also seem to be some dependence to but it is unclear from our results whether this effect influences the difference. Note that the larger the difference (i.e. 2 nd peak larger than the 1 st peak) the larger the . Therefore, it is important to consider these observations when performing the Franck-Condon analysis.  Estimation of Addition Energies. The addition energy, = ( ) − ( − 1), is defined as the energy difference between two MGNR charge states. We estimate an average = 156 meV from taking the energy from where the edges of the Coulomb diamonds meet and close, averaged over all the measured diamonds (Fig. S13c). Alternatively, the addition energy can be defined as the sum of the charging energy, , and the molecular electronic energy level spacing, Δ . We observe ≫ Δ and therefore = + Δ ≈ .
Estimation of MGNR Length. We assume that the MGNR is a rectangular quantum dot with width, = 1.12 nm, which length, , can be estimated as follows: 8 where is the elementary charge, 0 is the permittivity of free space, = 3.9 is the relative permittivity of the SiO2 layer 9 of thickness, ℎ = 300 nm is the distance between the gate electrode and the MGNR. Using Equation S4 we estimate = 33 nm for 2, and various lengths ( =28, 90, 40, 17, 108, 60, 50, 33) for the several MGNRs of 1.
Estimation of Lever Arm. The lever arm, , provides the relationship between and where Δ is the width of the Coulomb diamond in units of (i.e. volt). Using this method, we found that differs for different diamonds in a way that seems to depend on (Fig. S13d). is largest around small values of and declines symmetrically on either side of = 0. This feature indicates that either decreases or increases with increasing | |.
Calculation of Stability Diagram. To verify our conclusions, we used a quantum rateequation model to calculate a stability diagram following the procedure reported elsewhere 5,10 .
The model assumes a single vibrational mode with energy = 7 meV and a superohmic phonon bath. We assume that the single mode originates from the MGNR and that the phonon bath originates from the environment (dominantly the substrate) to which the MGNR is coupled. The expression for the current through a weakly coupled molecular junction is where ⁄ ⁄ denotes the diabatic rates of electron transfers at each electrode (S: source; D: drain) corresponding to a reduction (red) or an oxidation (ox) of the MGNR. Furthermore, for N/N+1-type transitions, the rates are given as follows: where Γ , is the electronic coupling between the MGNR and the source and drain leads, respectively. , ( ) is the Fermi-Dirac distribution in the respective electrode, is the chemical potential and ⁄ ( ) is the density of states of the MGNR written as: where = 1 for reduction and = −1 for oxidation, is the molecular energy level, is time, = 2ℏ (Γ S + Γ ) ⁄ is the lifetime of the MGNR electronic state and ( ) is the phononic correlation function: where is the single-mode contribution and ℎ is the contribution from the superohmic bath.
is the electron-vibration coupling strength, is the vibron frequency, is the Boltzmann constant and the spectral density, ( ), for the superohmic phonon bath with reorganisation energy, , and cut-off frequency, : (S12) Using this model we reproduced our experimental data by using the following parameters: Table S1: Parameters used with Eq. S5-S12 to calculate the stability diagram shown in Figure   S17b.
(mK) We assume that the molecular energy level, = − , shifts with bias and gate voltages. The calculated conductance = ⁄ was plotted as a function of and and is displayed next to an experimental stability diagram in Figure S17. The calculated and experimental stability diagrams match each other well. Importantly, the calculated diagram shows that the 'single-mode + environment' model reproduces the main features of our experimental data, i.e. conductance lines equally spaced by ≈ 7 meV, a significant contribution from the phonon bath and conductance asymmetry with bias. The calculated stability diagram contains additional conductance sidebands in the blocked regions that are not present in the experimental diagram.